What is the sum of infinity of the progression 1-x+x^2-x^3+…….
This is a geometric progression with first term 1 and common ratio -x. For a geometric progression to converge, its common ratio must be between -1 and 1. In this case, if x is between -1 and 1, the progression converges.
Using the formula for the sum of an infinite geometric progression, the sum of this series is:
S = 1 / (1 + x)
Therefore, if x is between -1 and 1, the sum of the infinite progression 1-x+x^2-x^3+…… is 1 / (1 + x).
To find the sum of an infinite geometric series, we need to check whether the common ratio is between -1 and 1. In this case, the common ratio is -x.
For the series to converge, we need to ensure that |-x| < 1, or that x > -1 and x < 1. Let's assume that x is within this range.
The sum of an infinite geometric series is given by the formula S = a / (1 - r), where 'S' is the sum of the series, 'a' is the first term, and 'r' is the common ratio.
In our case, the first term 'a' is 1, and the common ratio 'r' is -x. Thus, the sum of the series is:
S = 1 / (1 - (-x))
= 1 / (1 + x)
Therefore, the sum of the infinite series 1-x+x^2-x^3+... is 1 / (1 + x).